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Properties

Label 2-950-95.13-c1-0-5
Degree $2$
Conductor $950$
Sign $-0.772 - 0.634i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.996 + 0.0871i)2-s + (−1.70 + 0.794i)3-s + (0.984 + 0.173i)4-s + (−1.76 + 0.642i)6-s + (1.00 + 3.76i)7-s + (0.965 + 0.258i)8-s + (0.342 − 0.407i)9-s + (−1.56 − 2.71i)11-s + (−1.81 + 0.486i)12-s + (0.684 − 1.46i)13-s + (0.676 + 3.83i)14-s + (0.939 + 0.342i)16-s + (−0.551 + 6.30i)17-s + (0.376 − 0.376i)18-s + (−2.70 + 3.41i)19-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (−0.983 + 0.458i)3-s + (0.492 + 0.0868i)4-s + (−0.720 + 0.262i)6-s + (0.380 + 1.42i)7-s + (0.341 + 0.0915i)8-s + (0.114 − 0.135i)9-s + (−0.472 − 0.817i)11-s + (−0.524 + 0.140i)12-s + (0.189 − 0.407i)13-s + (0.180 + 1.02i)14-s + (0.234 + 0.0855i)16-s + (−0.133 + 1.52i)17-s + (0.0886 − 0.0886i)18-s + (−0.621 + 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.772 - 0.634i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.772 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424011 + 1.18489i\)
\(L(\frac12)\) \(\approx\) \(0.424011 + 1.18489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
5 \( 1 \)
19 \( 1 + (2.70 - 3.41i)T \)
good3 \( 1 + (1.70 - 0.794i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (-1.00 - 3.76i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.56 + 2.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.684 + 1.46i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.551 - 6.30i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-0.0240 + 0.0168i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (0.496 + 0.416i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (6.55 + 3.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.88 - 4.88i)T + 37iT^{2} \)
41 \( 1 + (2.29 - 6.29i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.28 - 3.26i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (13.3 - 1.16i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-0.168 - 0.241i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (-5.17 + 4.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.0829 - 0.470i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.685 - 7.83i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (5.21 - 0.918i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-2.70 - 5.79i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (-11.2 - 4.10i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (12.7 - 3.42i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.79 + 1.38i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.38 - 0.733i)T + (95.5 + 16.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64325658985033152571395002120, −9.775664782914324064172913208283, −8.315666954216892204435716282358, −8.190971619088983398693215890203, −6.37980430548575098067104972817, −5.85463574679299805814542810299, −5.35606203679710259371705824344, −4.37183294281053560048263724601, −3.18164249871294372624546855277, −1.93102744076290419334826719611, 0.50468050547885488767349366920, 1.95708780112062510594801648144, 3.48855089855697304459413762250, 4.66087990066841015711253306841, 5.11247935976965776141243180358, 6.34033254209485747390300214610, 7.17338167181453091975597721880, 7.39957549533471381154864064451, 8.932092017858959531218893328859, 10.02693393601787363088186179551

Graph of the $Z$-function along the critical line